Upon completing a program at one of these schools, do you then attempt to obtain an apprenticeship with a gunsmith to get some more working experience to add to a resume in before applying to major companies? Or do you just try to get whatever job you can to gain experience and then hope to get an opportunity with a larger company from there?

There's no one answer to this, it varies depending on your desires and any contacts you may have built up.

The SMART students start looking for a job about the time they start the last year. You should be looking HARD starting the last 6 months, and have at least one definite job offer lined up by graduation.

The schools offer job placement but don't put too much faith in that.
A graduate from Colorado or Trinidad is generally past the point of working as an apprentice unless you luck out and one of America's top custom gunsmiths will take you on.
Apprenticing with most ordinary gunsmiths will be of little real value other than giving you experience being in a working shop and dealing with the public.

Working for a gun company, a larger custom shop, for a company that does research and uses gunsmiths, or other companies who hire gunsmiths is probably the smartest way to start out.
That way, you spend 8 hours a day doing gunsmithing, make at lease decent money, can build up a reputation in the industry, build up a potential customer base, and buy the equipment over a period of time.

If the school offers business courses, like Trinidad does TAKE THEM. If not, go to night school.
Remember: If you open your own shop, you're NOT a gunsmith. You're a businessman who's business happens to be a gunsmith shop.
If you want to make it, you better keep that REAL straight in your mind.